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Approximating last-exit probabilities of a random walk, by application to conditional queue length moments within busy periods of M/GI/1 queues

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

D. J. Daley  and
L. D. Servi
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Abstract

Certain last-exit and first-passage probabilities for random walks are approximated via a heuristic method suggested by a ladder variable argument. They yield satisfactory approximations of the first- and second-order moments of the queue length within a busy period of an M/D/1 queue. The approximation is applied to the wider class of random walks that arise in studying M/GI/1 queues. For gamma-distributed service times the queue length distribution is independent of the arrival rate. For other distributions where the arrival rate affects the queue length distribution, we have to use conjugate distributions in order to exploit a local central limit property. The limit underlying the approximation has the nature of a Brownian excursion.

The source of the problem lies in recent queueing inference work; the connection with Takács' interests comes from both queueing theory and the ballot theorem.

Keywords

M/D/1 QUEUEM/GI/1 QUEUETABOO PROBABILITIESCONJUGATE DISTRIBUTIONS

MSC classification

Secondary: 60K25: Queueing theory
Type
Part 4 Random Walks
Information
Journal of Applied Probability , Volume 31 , Issue A , 1994 , pp. 251 - 267
Copyright
Copyright © Applied Probability Trust 1994 

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